Solve {ccc}{3}&{-1}&{-1}{-1}&{2}&{2}{-1}&{-1}&{3}

$$\left| \begin{array} { c c c } { 3 } & { - 1 } & { - 1 } \\ { - 1 } & { 2 } & { 2 } \\ { - 1 } & { - 1 } & { 3 } \end{array} \right|$$

Evaluate

$$20$$

Steps Using Sarrus' Rule

Find the determinant of the matrix using the method of diagonals.

$$det(\left(\begin{matrix}3&-1&-1\\-1&2&2\\-1&-1&3\end{matrix}\right))$$

Extend the original matrix by repeating the first two columns as the fourth and fifth columns.

$$\left(\begin{matrix}3&-1&-1&3&-1\\-1&2&2&-1&2\\-1&-1&3&-1&-1\end{matrix}\right)$$

Starting at the upper left entry, multiply down along the diagonals, and add the resulting products.

$$3\times 2\times 3-2\left(-1\right)-\left(-\left(-1\right)\right)=19$$

Starting at the lower left entry, multiply up along the diagonals, and add the resulting products.

$$-2\left(-1\right)-2\times 3+3\left(-1\right)\left(-1\right)=-1$$

Subtract the sum of the upward diagonal products from the sum of the downward diagonal products.

$$19-\left(-1\right)$$

Subtract -1 from 19.

$$20$$

Steps Using Expansion by Minors

Find the determinant of the matrix using the method of expansion by minors (also known as expansion by cofactors).

$$det(\left(\begin{matrix}3&-1&-1\\-1&2&2\\-1&-1&3\end{matrix}\right))$$

To expand by minors, multiply each element of the first row by its minor, which is the determinant of the 2\times 2 matrix created by deleting the row and column containing that element, then multiply by the element's position sign.

$$3det(\left(\begin{matrix}2&2\\-1&3\end{matrix}\right))-\left(-det(\left(\begin{matrix}-1&2\\-1&3\end{matrix}\right))\right)-det(\left(\begin{matrix}-1&2\\-1&-1\end{matrix}\right))$$

For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the determinant is ad-bc.

$$3\left(2\times 3-\left(-2\right)\right)-\left(-\left(-3-\left(-2\right)\right)\right)-\left(-\left(-1\right)-\left(-2\right)\right)$$

Simplify.

$$3\times 8-\left(-\left(-1\right)\right)-3$$

Add the terms to obtain the final result.

$$20$$

Factor

$$2^{2}\times 5$$